The Countering Iran's Destabilizing Activities Act, which was passed by the Senate a month ago, was held up in the House of Representatives after Republicans proposed including North Korea sanctions in the bill.
The House is set to vote on Tuesday on a package of bills on sanctions covering Russia, Iran and North Korea, according to House Majority Leader Kevin McCarthy's office. The measure will "hold them accountable for their dangerous actions," McCarthy said in a statement Saturday.
Under the proposed bill, Trump must submit to Congress a report on proposed actions that would "significantly alter" US foreign policy in connection with Russia, including easing sanctions or returning diplomatic properties in Maryland and New York that former President Barack Obama ordered vacated in December.
Congress would have at least 30 days to hold hearings and then vote to uphold or reject Trump's proposed changes.
Many lawmakers hope the bill will send a message to Trump to keep a strong line against Russia.
Trump, who met Putin at the Group of 20 summit in Hamburg earlier this month and said it was an "honor" to meet him, has been criticized for seeking to reset US-Russian relations. His administration has been bogged down by ongoing investigations of possible ties between his 2016 campaign and Russia.
With the bill, Republicans and Democrats are seeking to punish Russia for its 2014 annexation of Crimea, a peninsula belong to Ukraine, and for meddling in the 2016 presidential election.
Putin has denied any meddling in the US democratic process last year. Trump has said that his campaign did not collude with Russia.
House Democratic leader Nancy Pelosi said Russia's "outrageous and unacceptable" behavior in the 2016 US election and in Europe "demand that we have strong statutory sanctions enacted as soon as possible."
Senior Republican lawmakers did not immediately comment on the latest bill.
In Brussels, the European Union sounded an alarm about the US moves to step up sanctions on Russia, urging Washington to coordinate with its Group of 7 partners.
The European Commission, the EU's executive body, warned of possibly "wide and indiscriminate" "unintended consequences," notably on the EU's efforts to diversify energy sources away from Russia.
McCarthy and Ed Royce, chair of the House Foreign Affairs Committee, said in a statement the revised bill helps "bolster the energy security of our European allies by maintaining their access to key energy resources outside of Russia."
Senate Democratic leader Chuck Schumer said a strong sanctions bill "is essential," and said in a statement that he expects "the House and Senate will act on this legislation promptly, on a broad bipartisan basis."
Senator Ben Cardin, the top Democrat on the Senate Foreign Relations Committee, said the agreement was reached after "intense negotiations."
"A nearly united Congress is poised to send President Putin a clear message on behalf of the American people and our allies, and we need President Trump to help us deliver that message," he said in a statement.
(Reporting by Lesley Wroughton and David Shepardson in Washington; Editing by Leslie Adler)